## Summary and Analysis of Extension Program Evaluation in R

Salvatore S. Mangiafico

# Correlation and Linear Regression

Correlation and linear regression each explore the relationship between two quantitative variables.  Both are very common analyses.

Correlation determines if one variable varies systematically as another variable changes.  It does not specify that one variable is the dependent variable and the other is the independent variable.  Often, it is useful to look at which variables are correlated to others in a data set, and it is especially useful to see which variables correlate to a particular variable of interest.

The three forms of correlation presented here are Pearson, Kendall, and Spearman.  The test determining the p-value for Pearson correlation is a parametric test that assumes that data are bivariate normal.  Kendall and Spearman correlation use nonparametric tests.

In contrast, linear regression specifies one variable as the independent variable and another as the dependent variable.  The resultant model relates the variables with a linear relationship.

The tests associated with linear regression are parametric and assume normality, homoscedasticity, and independence of residuals, as well as a linear relationship between the two variables.

##### Appropriate data

•  For Pearson correlation, two interval/ratio variables.  Together the data in the variables are bivariate normal.  The relationship between the two variables is linear.  Outliers can detrimentally affect results.

•  For Kendall correlation, two variables of interval/ratio or ordinal type.

•  For Spearman correlation, two variables of interval/ratio or ordinal type.

•  For linear regression, two interval/ratio variables.  The relationship between the two variables is linear.  Residuals are normal, independent, and homoscedastic.  Outliers can affect the results unless robust methods are used.

##### Hypotheses

•  For correlation, null hypothesis:  The correlation coefficient (r, tau, or rho) for the sampled population is zero.  Or, there is no correlation between the two variables.

•  For linear regression, null hypothesis:  The slope of the fit line for the sampled population is zero.  Or, there is no linear relationship between the two variables.

##### Interpretation

•  For correlation, reporting significant results as “Variable A was significantly correlated to Variable B” is acceptable.  Alternatively, “A significant correlation between Variable A and Variable B was found.”  Or, “Variables A, B, and C were significantly correlated.”

•  For linear regression, reporting significant results as “Variable A was significantly linearly related to Variable B” is acceptable.  Alternatively, “A significant linear regression between Variable A and Variable B was found.”

### Packages used in this chapter

The packages used in this chapter include:

•  psych

•  PerformanceAnalytics

•  ggplot2

•  rcompanion

The following commands will install these packages if they are not already installed:

if(!require(psych)){install.packages("psych")}
if(!require(PerformanceAnalytics)){install.packages("PerformanceAnalytics")}
if(!require(ggplot2)){install.packages("ggplot2")}
if(!require(rcompanion)){install.packages("rcompanion")}

### Examples for correlation and linear regression

Brendon Small and company recorded several measurements for students in their classes related to their nutrition education program: Grade, Weight in kilograms, intake of Calories per day, daily Sodium intake in milligrams, and Score on the assessment of knowledge gain.

Input = ("
Instructor       Grade   Weight  Calories Sodium  Score
'Brendon Small'     6      43     2069    1287      77
'Brendon Small'     6      41     1990    1164      76
'Brendon Small'     6      40     1975    1177      76
'Brendon Small'     6      44     2116    1262      84
'Brendon Small'     6      45     2161    1271      86
'Brendon Small'     6      44     2091    1222      87
'Brendon Small'     6      48     2236    1377      90
'Brendon Small'     6      47     2198    1288      78
'Brendon Small'     6      46     2190    1284      89
'Jason Penopolis'   7      45     2134    1262      76
'Jason Penopolis'   7      45     2128    1281      80
'Jason Penopolis'   7      46     2190    1305      84
'Jason Penopolis'   7      43     2070    1199      68
'Jason Penopolis'   7      48     2266    1368      85
'Jason Penopolis'   7      47     2216    1340      76
'Jason Penopolis'   7      47     2203    1273      69
'Jason Penopolis'   7      43     2040    1277      86
'Jason Penopolis'   7      48     2248    1329      81
'Melissa Robins'    8      48     2265    1361      67
'Melissa Robins'    8      46     2184    1268      68
'Melissa Robins'    8      53     2441    1380      66
'Melissa Robins'    8      48     2234    1386      65
'Melissa Robins'    8      52     2403    1408      70
'Melissa Robins'    8      53     2438    1380      83
'Melissa Robins'    8      52     2360    1378      74
'Melissa Robins'    8      51     2344    1413      65
'Melissa Robins'    8      51     2351    1400      68
'Paula Small'       9      52     2390    1412      78
'Paula Small'       9      54     2470    1422      62
'Paula Small'       9      49     2280    1382      61
'Paula Small'       9      50     2308    1410      72
'Paula Small'       9      55     2505    1410      80
'Paula Small'       9      52     2409    1382      60
'Paula Small'       9      53     2431    1422      70
'Paula Small'       9      56     2523    1388      79
'Paula Small'       9      50     2315    1404      71
'Coach McGuirk'    10      52     2406    1420      68
'Coach McGuirk'    10      58     2699    1405      65
'Coach McGuirk'    10      57     2571    1400      64
'Coach McGuirk'    10      52     2394    1420      69
'Coach McGuirk'    10      55     2518    1379      70
'Coach McGuirk'    10      52     2379    1393      61
'Coach McGuirk'    10      59     2636    1417      70
'Coach McGuirk'    10      54     2465    1414      59
'Coach McGuirk'    10      54     2479    1383      61
")

###  Order factors by the order in data frame
###  Otherwise, R will alphabetize them

Data\$Instructor = factor(Data\$Instructor,
levels=unique(Data\$Instructor))

###  Check the data frame

library(psych)

str(Data)

summary(Data)

### Remove unnecessary objects

rm(Input)

#### Visualizing correlated variables

##### Correlation does not have independent and dependent variables

Notice in the formulae for the pairs and cor.test functions, that all variables are named to the right of the ~.  One reason for this is that correlation is measured between two variables without assuming that one variable is the dependent variable and one is the independent variable.  By necessity, one variable must be plotted on the x-axis and one on the y-axis, but their placement is arbitrary.

##### Multiple correlation

The pairs function can plot multiple numeric or integer variables on a single plot to look for correlations among the variables.

pairs(data=Data,
~ Grade + Weight + Calories + Sodium + Score)

The corr.test function in the psych package can be used in a similar manner, with the output being a table of correlation coefficients and a table of p-values.  p-values can be adjusted with the adjust= option.  Options for correlation methods are “pearson”, “kendall”, and “spearman”.  The function can produce confidence intervals for the correlation coefficients, but I recommend using the cor.ci function in the psych package for this task (not shown here).

The corr.test function requires that the data frame contain only numeric or integer variables, so we will first create a new data frame called Data.num containing only the numeric and integer variables.

Data.num = Data[c("Grade", "Weight", "Calories", "Sodium", "Score")]

library(psych)

corr.test(Data.num,
use    = "pairwise",
method = "pearson",

Correlation matrix

Grade     1.00   0.85     0.85   0.79 -0.70
Weight    0.85   1.00     0.99   0.87 -0.48
Calories  0.85   0.99     1.00   0.85 -0.48
Sodium    0.79   0.87     0.85   1.00 -0.45
Score    -0.70  -0.48    -0.48  -0.45  1.00

Sample Size
[1] 45

A useful plot of histograms, correlations, and correlation coefficients can be produced with the chart.Correlation function in the PerformanceAnalytics package.

In the output, the numbers represent the correlation coefficients (r, tau, or rho, depending on whether Pearson, Kendall, or Spearman correlation is selected, respectively).  The stars represent the p-value of the correlation:

*     p < 0.05
**    p < 0.01
***   p < 0.001

The function requires that the data frame contain only numeric or integer variables, so we will first create a new data frame called Data.num containing only the numeric and integer variables.

Data.num = Data[c("Grade", "Weight", "Calories", "Sodium", "Score")]

library(PerformanceAnalytics)

chart.Correlation(Data.num,
method="pearson",
histogram=TRUE,
pch=16)

#### Effect size statistics

##### r, rho, and tau

The statistics r, rho, and tau are used as effect sizes for Pearson, Spearman, and Kendall regression, respectively.  These statistics vary from –1 to 1, with 0 indicating no correlation, 1 indicating a perfect positive correlation, and –1 indicating a perfect negative correlation.  Like other effect size statistics, these statistics are not affected by sample size.

Interpretation of effect sizes necessarily varies by discipline and the expectations of the experiment.  They should not be considered universal.  An interpretation of r is given by Cohen (1988).  It is probably reasonable to use similar interpretations for rho and tau.

 small medium large r 0.10  – < 0.30 0.30  – < 0.50 ≥ 0.50

________________________________

##### r-squared

For linear regression, r-squared is used as an effect size statistic.  It indicates the proportion of the variability in the dependent variable that is explained by model.  That is, an r-squared of 0.60 indicates that 60% of the variability in the dependent variable is explained by the model.

#### Pearson correlation

The test used for Pearson correlation is a parametric analysis that requires that the relationship between the variables is linear, and that the data be bivariate normal.  Variables should be interval/ratio.  The test is sensitive to outliers.

The correlation coefficient, r, can range from +1 to –1, with +1 being a perfect positive correlation and –1 being a perfect negative correlation.  An r of 0 represents no correlation whatsoever.  The hypothesis test determines if the r value is significantly different from 0.

As an example, we’ll plot Sodium vs. Calories, and use the cor.test function to test the correlation of these two variables.

##### Simple plot of the data

Note that the relationship between the y and x variables is not particularly linear.

plot(Sodium ~ Calories,
data=Data,
pch=16,
xlab = "Calories",
ylab = "Sodium")

##### Pearson correlation

Note that the results report the p-value for the hypothesis test as well as the r value, written as cor, 0.849.

cor.test( ~ Sodium + Calories,
data=Data,
method = "pearson")

Pearson's product-moment correlation

t = 10.534, df = 43, p-value = 1.737e-13
alternative hypothesis: true correlation is not equal to 0

95 percent confidence interval:
0.7397691 0.9145785

sample estimates:
cor
0.8489548

###### Plot residuals

It’s not a bad idea to look at the residuals from Pearson correlation to be sure the data meet the assumption of bivariate normality.  Unfortunately, the cor.test function doesn’t supply residuals.  One solution is to use the lm function, which actually redoes the analysis as a linear regression.

model = lm(Sodium ~ Calories,
data = Data)

x = residuals(model)

library(rcompanion)

plotNormalHistogram(x)

#### Kendall correlation

Kendall correlation is considered a nonparametric analysis.  It is a rank-based test that does not require assumptions about the distribution of the data.  Variables can be interval/ratio or ordinal.

The correlation coefficient from the test is tau, which can range from +1 to –1, with +1 being a perfect positive correlation and –1 being a perfect negative correlation.  A tau of 0 represents no correlation whatsoever.  The hypothesis test determines if the tau value is significantly different from 0.

As a technical note, the cor.test function in R calculates tau-b, which handles ties in ranks well.

The test is relatively robust to outliers in the data.  The test is sometimes cited for being reliable when there are small number of samples or when there are many ties in ranks.

##### Simple plot of the data

(See Pearson correlation above.)

##### Kendall correlation

Note that the results report the p-value for the hypothesis test as well as the tau value, written as tau, 0.649.

cor.test( ~ Sodium + Calories,
data=Data,
method = "kendall")

Kendall's rank correlation tau

z = 6.2631, p-value = 3.774e-10
alternative hypothesis: true tau is not equal to 0

sample estimates:
tau
0.6490902

#### Spearman correlation

Spearman correlation is considered a nonparametric analysis.  It is a rank-based test that does not require assumptions about the distribution of the data.  Variables can be interval/ratio or ordinal.

The correlation coefficient from the test, rho, can range from +1 to –1, with +1 being a perfect positive correlation and –1 being a perfect negative correlation.  A rho of 0 represents no correlation whatsoever. The hypothesis test determines if the rho value is significantly different from 0.

Spearman correlation is probably most often used with ordinal data.  It tests for a monotonic relationship between the variables.  It is relatively robust to outliers in the data.

##### Simple plot of the data

(See Pearson correlation above.)

##### Spearman correlation

Note that the results report the p-value for the hypothesis test as well as the rho value, written as rho, 0.820.

cor.test( ~ Sodium + Calories,
data=Data,
method = "spearman")

Spearman's rank correlation rho

S = 2729.7, p-value = 5.443e-12
alternative hypothesis: true rho is not equal to 0

sample estimates:
rho
0.8201766

#### Linear regression

Linear regression is a very common approach to model the relationship between two interval/ratio variables.  The method assumes that there is a linear relationship between the dependent variable and the independent variable, and finds a best fit model for this relationship.

###### Dependent and Independent variables

When plotted, the dependent variable is usually placed on the y-axis, and the independent variable is usually placed in the x-axis.

###### Interpretation of coefficients

The outcome of linear regression includes estimating the intercept and the slope of the linear model.  Linear regression can then be used as a predictive model, whereby the model can be used to predict a y value for any given x.  In practice, the model shouldn’t be used to predict values beyond the range of the x values used to develop the model.

###### Multiple, nominal, and ordinal independent variables

If there are multiple independent variables of interval/ratio type in the model, then linear regression expands to multiple regression.  The polynomial regression example in this chapter is a form of multiple regression.

If the independent variable were of nominal type, then the linear regression would become a one-way analysis of variance.

Handling independent variables of ordinal type can be complicated.  Often they are treated as either nominal type or interval/ratio type, although there are drawbacks to each approach.

###### Assumptions

Linear regression assumes a linear relationship between the two variables, normality of the residuals, independence of the residuals, and homoscedasticity of residuals.

###### Note on writing r-squared

For bivariate linear regression, the r-squared value often uses a lower case r; however, some authors prefer to use a capital R.  For multiple regression, the R in the R-squared value is usually capitalized.  The name of the statistic may be written out as “r-squared” for convenience, or as r2.

##### Define model, and produce model coefficients, p-value, and r-squared value

Linear regression can be performed with the lm function, which was the same function we used for analysis of variance.

The summary function for lm model objects includes estimates for model parameters (intercept and slope), as well as an r-squared value for the model and p-value for the model.

Note that even for bivariate regression, the output calls the r-squared value “Multiple R-squared”.

model = lm(Sodium ~ Calories,
data = Data)

summary(model)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 519.07547   78.78211   6.589 5.09e-08 ***
Calories      0.35909    0.03409  10.534 1.74e-13 ***

Residual standard error: 38.89 on 43 degrees of freedom
Multiple R-squared:  0.7207,  Adjusted R-squared:  0.7142
F-statistic:   111 on 1 and 43 DF,  p-value: 1.737e-13

##### Plot data with best fit line

plot(Sodium ~ Calories,
data=Data,
pch=16,
xlab = "Calories",
ylab = "Sodium")

abline(model,
col = "blue",
lwd = 2)

There is a clear nonlinearity to the data, suggesting that the simple linear model is not the best fit.  This nonlinearity is apparent in the plot of residuals vs. fitted values as well.

The next chapter will include fitting linear plateau, quadratic plateau, and a curvilinear models with this data.

The following section will attempt to improve the model fit by adding polynomial terms.

##### Plots of residuals

x = residuals(model)

library(rcompanion)

plotNormalHistogram(x)

plot(fitted(model),
residuals(model))

#### Polynomial regression

Polynomial regression adds additional terms to the model, so that the terms include some set of the linear, quadratic, cubic, and quartic, etc., forms of the independent variable.  These terms are the independent variable, the square of the independent variable, the cube of the independent variable, and so on.

##### Create polynomial terms in the data frame

Because the variable Calories is an integer variable, we will need to convert it to a numeric variable.  Otherwise R will produce errors when we try to square and cube the values of the variable.

Our new variables will be Calories2 for the square of Calories, Calories3 for the cube of Calories, and so on.

Data\$Calories = as.numeric(Data\$Calories)

Data\$Calories2 = Data\$Calories * Data\$Calories
Data\$Calories3 = Data\$Calories * Data\$Calories * Data\$Calories
Data\$Calories4 = Data\$Calories * Data\$Calories * Data\$Calories * Data\$Calories

##### Define models and determine best model

Chances are that we will not need all of the polynomial terms to adequately model our data.  One approach to choosing the best model is to construct several models with increasing numbers of polynomial terms, and then use a model selection criterion like AIC, AICc, or BIC to choose the best one.  These criteria balance the goodness-of-fit of each model versus its complexity.  That is, the goal is to find a model which adequately explains the data without having too many terms.

We can use the compareLM function to list AIC, AICc, and BIC for a series of models.

model.1 = lm(Sodium ~ Calories,
data = Data)

model.2 = lm(Sodium ~ Calories + Calories2,
data = Data)

model.3 = lm(Sodium ~ Calories + Calories2 + Calories3,
data = Data)

model.4 = lm(Sodium ~ Calories + Calories2 + Calories3 + Calories4,
data = Data)

library(rcompanion)

compareLM(model.1, model.2, model.3, model.4)

\$Models
Formula
1 "Sodium ~ Calories"
2 "Sodium ~ Calories + Calories2"
3 "Sodium ~ Calories + Calories2 + Calories3"
4 "Sodium ~ Calories + Calories2 + Calories3 + Calories4"

\$Fit.criteria
Rank Df.res   AIC  AICc   BIC R.squared Adj.R.sq   p.value Shapiro.W Shapiro.p
1    2     43 461.1 461.7 466.5    0.7207   0.7142 1.737e-13    0.9696    0.2800
2    3     42 431.4 432.4 438.6    0.8621   0.8556 8.487e-19    0.9791    0.5839
3    4     41 433.2 434.7 442.2    0.8627   0.8526 1.025e-17    0.9749    0.4288
4    5     40 428.5 430.8 439.4    0.8815   0.8696 5.566e-18    0.9625    0.1518

If we use BIC as our model fit criterion, we would choose model.2 as the best model for these data, since it had the lowest BIC.  AIC or AICc could have been used as criteria instead.

BIC tends to penalize models more than the other criteria for having additional parameters, so it will tend to choose models with fewer terms.

There is not generally accepted advice as to which model fitting criterion to use.  My advice might be to use BIC when a more parsimonious model (one with fewer terms) is a priority, and in other cases use AICc.

##### For final model, produce model coefficients, p-value, and R-squared value

summary(model.2)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.707e+03  6.463e+02  -5.736 9.53e-07 ***
Calories     4.034e+00  5.604e-01   7.198 7.58e-09 ***
Calories2   -7.946e-04  1.211e-04  -6.563 6.14e-08 ***

Residual standard error: 27.65 on 42 degrees of freedom
Multiple R-squared:  0.8621,  Adjusted R-squared:  0.8556
F-statistic: 131.3 on 2 and 42 DF,  p-value: < 2.2e-16

##### Note on retaining lower-order effects

In polynomial regression, we typically keep all lower order effects of any higher order effects we include.  So, in this case, if the final model included Calories2, but the effect of Calories were not significant, we would still include Calories in the model since we are including Calories2.

##### Plot data with best fit line

For bivariate data, the function plotPredy will plot the data and the predicted line for the model.  It also works for polynomial functions, if the order option is changed.

library(rcompanion)

plotPredy(data  = Data,
y     = Sodium,
x     = Calories,
x2    = Calories2,
model = model.2,
order = 2,
xlab  = "Calories per day",
ylab  = "Sodium intake per day")

##### Plot of best fit line with confidence interval

For polynomial regression, the ggplot function can plot the best-fit curve with the confidence interval of the curve shaded.

library(ggplot2)

ggplot(Data,
aes(x = Calories,
y = Sodium)) +
geom_point() +
geom_smooth(method  = "lm",
formula = y ~ poly(x, 2, raw=TRUE), ### polynomial of order 2

se      = TRUE)

##### Plots of residuals

x = residuals(model.2)

library(rcompanion)

plotNormalHistogram(x)

plot(fitted(model.2),
residuals(model.2))

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### References

Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences, 2nd Edition. Routledge.

Correlation and Linear Regression in Mangiafico, S.S. 2015. An R Companion for the Handbook of Biological Statistics, version 1.09. rcompanion.org/rcompanion/e_01.html.

Multiple Regression in Mangiafico, S.S. 2015. An R Companion for the Handbook of Biological Statistics, version 1.09. rcompanion.org/rcompanion/e_05.html.

### Exercises U

1. Consider the data from Brendon, Jason, Melissa, Paula, and McGuirk.  Report for each answer, indicate how you know, when appropriate, by reporting the values of the statistic you are using or other information you used.

a.  Which two variables are the most strongly correlated?

b.  Which two variables are the least strongly correlated?

c.  Are there any pairs of variables that are statistically uncorrelated?  Which?

d.  Name a pair of variables that is positively correlated.

e.  Name a pair of variables that is negatively correlated.

f.  Is Sodium significantly correlated with Calories?

g.  By linear regression, is there a significant linear relationship of Sodium vs. Calories?

h.  Does the quadratic polynomial model fit the Sodium vs. Calories data better than the linear model?  Consider the p-value, the r-squared value, the range of values for each of Sodium and Calories, and your practical conclusions.

2.  As part of a professional skills program, a 4-H club tests its members for typing proficiency (Words.per.minute), Proofreading skill, proficiency with using a Spreadsheet, and acumen in Statistics.

'Dr. Katz'  6     35               53           75           61
'Dr. Katz'  6     50               77           24           51
'Dr. Katz'  6     55               71           62           55
'Dr. Katz'  6     60               78           27           91
'Dr. Katz'  6     65               84           44           95
'Dr. Katz'  6     60               79           38           50
'Dr. Katz'  6     70               96           12           94
'Dr. Katz'  6     55               61           55           76
'Dr. Katz'  6     45               73           59           75
'Dr. Katz'  6     55               75           55           80
'Dr. Katz'  6     60               85           35           84
'Dr. Katz'  6     45               61           49           80
'Laura'     7     55               59           79           57
'Laura'     7     60               60           60           60
'Laura'     7     75               90           19           64
'Laura'     7     65               87           32           65
'Laura'     7     60               70           33           94
'Laura'     7     70               84           27           54
'Laura'     7     75               87           24           59
'Laura'     7     70               97           38           74
'Laura'     7     65               86           30           52
'Laura'     7     72               91           36           66
'Laura'     7     73               88           20           57
'Laura'     7     65               86           19           71
'Ben Katz'  8     55               84           20           76
'Ben Katz'  8     55               63           44           94
'Ben Katz'  8     70               95           31           88
'Ben Katz'  8     55               63           69           93
'Ben Katz'  8     65               65           47           70
'Ben Katz'  8     60               61           63           92
'Ben Katz'  8     70               80           35           60
'Ben Katz'  8     60               88           38           58
'Ben Katz'  8     60               71           65           99
'Ben Katz'  8     62               78           46           54
'Ben Katz'  8     63               89           17           60
'Ben Katz'  8     65               75           33           77

For each of the following, answer the question, and show the output from the analyses you used to answer the question.  Where relevant, indicate how you know.

a.  Which two variables are the most strongly correlated?

b.  Name a pair of variables that are statistically uncorrelated.

c.  Name a pair of variables that is positively correlated.

d.  Name a pair of variables that is negatively correlated.

i.  What is the value of the correlation coefficient r for this correlation?

ii.  What is the value of tau

iii.  What is the value of rho?

f.  Conduct a linear regression of Proofreading vs. Words.per.minute.

i.  What is the p-value for this model?

ii.  What is the r-squared value?

iii.  Do the residuals suggest that the linear regression model is an appropriate model?

iv.  What can you conclude about the results of the linear regression?  Consider the p-value, the r-squared value, the range of values for each of Proofreading and Words.per.minute, and your practical conclusions.